1 Answer
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There are two parts to the game's progress. There's the story progress, which is stored in the game's save file, and there's character progress. Starting one over would not affect the other.

To start the game from scratch, you can either:a. Choose a different save slot upon starting the game (I know the PS3 version allows three save slots, but I don't know about other versions).orb. Delete your save through the console.
The game saves includes all completed levels, objectives and collectibles (legendary treasures, scrolls, hat and soul gems). Like I said, it doesn't include character progression.

To start a character over, you need to reset it. To do so, place it on your portal after loading a save and enter the character menu (- on the Wii, Select on the PS3 and Back on the Xbox 360) and choose to reset the progress. This will set the character back at level 1 without any gems, upgrades or hats.
Please note this is irreversible, and will affect all versions and saves of the game playing with that character.

To sum it up, if you want to completely reset the game to it's initial state, you'll need to reset all characters in the method specified and delete the save.